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Progress in Mineral Processing Technology Demirel amp Ersaym (eds) copy 1994 Balkema Rotterdam ISBN 90 5410 513 5
Effects of some variables on the cumulative basis kinetic model parameters
in ball mill grinding
Selahattin Ana~ Etibank Project and Implementation Department Ankara Turkey
S Levent Ergtin amp Birol Sonmez University ofHacettepe Department ofMining Engineering Ankara Turkey
ABSTRACT In this study the effects of ball charge powder filling mill speed and ball size on model parameters ofcumulative basis kinetic model ofball mill grinding C and n were examined The standard Bond mill was used in the experiments Mogul lead-zinc ore was used as the material The results indicated that parameter n could be considered a material dependent parameter It remained constant at acceptable confidence level and the changes in operating conditions mainly affected the value of parameter C During the range studied it is found that optimum values for ball charge and ball size exist for maximizing the value of C while it asymptotically increased with increasMg critical speed and decrease with increasing powder filling The implications of the findings for efficient grinding and scale-up are discussed
11gt
1 INTRODUCTION
Mathematical modelling of grinding operations have been studied extensively in the literature The wellshyknown models are matrix model (Lynch 1977) sizeshymass balance kinetic model (Austin et aI 1984) and population balance kinetic model (Herbst and Fuerstenau 1980) and multi-segment kinetic model (Lynch et aI 1986) Despite their robustness and physical significance these models have some experimental and computational difficulties Cumulative basis kinetic model was selected over more complex models in many applications (Ramirez-Castro and Finch 1980 1981 Laplante etal 1987 Fidan 1990 Ersaym etal 1993) due to its experimental and computational easiness The results were rather successful and quite comparable with complex models
Although cumulative basis kinetic model have been used for simulation purposes the interaction between operating conditions and model parameters have not been fully described yet Olsen and Krogh (1972) have att~mpted to develop a mathematical expression for this but their experimental data are limited that only the effects of some operating variables on rate parameters were investigated In this study the effects of some important
variables in batch grinding on model parameters of cumulative basis kinetic model were investigated
2 EXPERIMENTAL STUDIES
Grinding tests were carried out in a purpose designed laboratory mill which has the same dimensions with the Bond mill (ie305x305 cm) The mill speed could be adjusted to the range among 25-250 rpm Mogul lead-zinc ore from Ireland was used as test material The material was first crushed to -336mm then samples were taken from the lot by riffle sampler The dry grinding tests were carried out for 124 and 8 minute The experimental range studied are presented in the
Table 1
Table L The experimental range studied and reference values
Range Reference ball fillingI 20-44 44 mill speed2 44-88 88 powder filling3 9-57 18 ball size (cm) 127-381 254
defined as (volume of balls + void volume between balls) I mill volume-() 2 defined as ~of critical speed 3 defined as ofvoid volume between balls filled by material
All the tests werd carried out varying one parameter at a time keeping the other parameters fixed at their reference values The reference values were 44
533
ball filling 875 of critical speed 18 powder filling and 254 cm diameter balls
After each grinding period the whole load in the mill was emptied and a representative sample was taken for sieve analysis Then the material retained on the screens were combined with the rest of mill discharge as a feed for next grinding period After sieve analysis model parameters cumulative basis kinetic model were determined
For the purposes of comparison and scale-up another test was carried out under the standard Bond test conditions (Bergstrom 1985) and the two ball mill operating parallel in Mogul plant were sampled After mass balancing and adjusting the raw data based on plug flow assumption the retention time of 591 minute was estimated for ball mill From the combined feed (fresh feed + cyclone undersize) and mill discharge size distributions the rate parameters were calculated by using the estimated retention time The model parameters C and n of the mills were then determined
Cumulative basis kinetic model
(1)Wi(t) = Wi(O)middote-kit ki=CxP (2)
cumulative oversize of size i after minute grinding
Wi(O)
Wi(t)
cumulative oversize of size i in the feed ki rate parameter (min-I) t grinding time (min) Xi particle size (Ilm) Cn model parameters
From size distribution data first rate parameter k of each size fraction and then using Equation 2 C and n parameters were determined A non-linear regression technique was used for the calculation of model parameters
3 RESULTS AND DISCUSSION
After the experiments the experimental data showed a quite good fit to both Equation 1 and 2 except the results of the test conducted with 127 cm balls This may be due to insufficient energy exerted to the particles from 127 cm diameter balls Effects of operational conditions on k parameters are shown at Figure 1-4
As can be seen from the figures the effects of operating conditions on rate parameter k were more pronounced at coarser size range Rate parameters decreased with increasing powder filling while they increased asymptotically with increasing mill speed It also appeared that there exist optimum ball size and ball filling for obtaining maximum efficiency
534
---+--425flIl 08 --IJ-300flIl
88GiSpeed middotmiddotmiddotmiddotjmiddotmiddotmiddotmiddot212flIl06 _ __ 150flIl
x04 ___ l06flIl
middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot75flIl02 I ~ltoo~ I --+--53flIl o I I I
5 10 15 20 25 30 35 40 powler filling
Figure 1 Effect of powder filling on rate parameters of different sizes (88 critical speed)
02 ----------- --+-425flIl
oos
44Gi~ ---- - lOOflIl 015 ~
~~ l 212flIl
OJ
~~ _middot-X-middot-lSOfID
--I--l~~~~ ---e--75flIl
--+-53flIl
O+I---+----r--~
15 30 45 ro jXJoOOrfilling
Figure 2 Effect of powder filling on rate parameters of different sizes (44 critical speed)
05
1 ---+-- 425fJ11
--IJ-300fJ11 04 ------~--
j 212fJ11 ~--03 ~l
-middot-)E-middot_middot 150fJ11I E--------_x
__- 106fJ11 ~=CCCl emiddotmiddot75fJ11
o -+-53fJ11
15 25 35 45 ballfilling
Figure 3 Effect of ball filling on rate parameters of different sizes
c 6 g
a ~
001 lts I I
Immmmll IOlg~~~
10 particle sire (pm) 1000
Figure 6 Relationship between rate parameter and particle size for different conditions
100
Candn
o constantnfmiddot bull _J II70 ]60 ]50 1
140 30 1
20 10
o I
~V
~ W~SWlln~
I~ 0
0 20 40 60 80 1~1 observed cumulative ovemizc(
Figure 7 Comparison of fit obtained by using C and n and constant n
001 poMerlilling
o~ ~ bull ~scil 0006 bull 4375Csi
----~s CS 0004 Imiddot middotmiddot 4375 cs1
bull0002 ~ --bull o +-1---+---+----1------+-----1----1
o 10 20 30 40 50 00
Figure 8 Effect of powder filling on C parameter
536
~~ r----------------------------- 0007
~006
0005 ~-~ 0004
0003
0002 +--------+--------+------------1 15 25 35 45
Figure 9 Effect ofball filling on C parameter
~006
ball size 0005
10004
I ~003
I 0002
I0001 1 2 3
Figure 10 Effect ofbaIl size on C parameter
0005 ~I
0004
Trill speed 0003
OOOZ+I----~------~----+_----~----~
40 50 00 70 80
Figure 11 Effect ofmill speed on C parameter
4
05
Q4
03 - Q2
Ql
o +---~~~~---+---+--~ 1 U 2 ~ 3 ~ 4
w~
Figure 4 Effect of ball size on rate parameters of different sizes
-+-~ 425flIl 04
---middot300flIl
middotmiddotmiddotmiddotmiddottmiddotmiddotmiddotmiddotmiddotmiddot 212flIl
-_- 150flIl 02
03 ~--------
t~ _-- l06flIl
bullbullbull 15flIlOJ -middot+-53flIl
I I I Io 40 50 ro 70 ID lt0
niIlspeed
Figure 5 Effect of mill speed on rate parameters of different sizes
The model parameters calculated from rate parameters are presented at Table 2 A close examination of the values of n obtained under different laboratory conditions (excluding the test under Bond conditions) revealed that n was almost constant and the changes in conditions were mainly reflected on the values of parameter C This can also be seen from Figure 6 where the slopes of lines were nearly the same indicating constant n There was only one test at which n showed large deviation This was obtained when ball size of 127 cm was used As explained earlier inefficient grinding with this size of balls was observed Excluding this test it may be assumed that n could be considered as a material dependent parameter This would mean that the changes on grinding conditions will be reflected on the values of parameter C
535
Table 2 C and n parameters ofthe tests
Cx163 n powder filling () at 875 cs 89 758 078 178 559 070 356 lAO 074 at 43 cs 178 254 071 356 283 064 572 142 064 mill speed ( of cs) 438 254 071 625 221 085 875 559 070 ball filling () 179 219 077 265 412 079 441 559 070 ball size (cm) 127 012 040 254 559 070 381 250 081 Bond Conditions 324 074
To check the validity of this assumption the mean value of n (excluding the extreme result) was first used in the re-calculations ofparameter C Then the predicted size distributions were calculated by using the new model parameters As is shown in Figure 7 both approaches provided similar fit to the data Thereforeit was concluded that under the laboratory conditions studied the model parameter n was reasonably constant indicating that it may be a material dependent parameter
The relationships between the variables studied and the re-calculated (for n=07316) C parameter are presented in Figure 8-11 The trends were similar to those obtained for the rate parameters Of the variables studied the most significant effect was observed for the powder filling Le C decreased sharply with its increase However the curvature gives the impression that further increase would not have such sharp effect Perhaps it would stay almost constant after a certain level Nevertheless this implies that plant scale grinding is performed at inefficient conditions since mills are normally operated at 100-120 powder filling On the other hand the optimum obtained for ball filling is around the normal operating conditions and an expected relationship was observed between C and critical speed It may be suggested the optimum ball size could vary with the feed size ofthe material
Although the plot of log k vs log particle size showed relatively larger deviation from linearity for the test conducted under Bond conditions (thick solid line in Figure 6) the value of n calculated by non-linear regression was almost the same as the mean values ofthe other laboratory tests Since there is only one set of data it would be very much arguable to suggest that the mixture of ball might be the cause of the deviation from linearity From the examination of rate parameters of individual sizes It appears that deviation mainly occurs due to inefficient grinding in fine sizes Although not so much pronounced similar deviations are also discernible in the parameters of the other tests The value of C obtained from the test under Bond
conditions is within the range studied but it did not exactly correspond to the most similar test conditions with one size balls It is slightly lower than the expected The model parameters calculated from the plant data is given Table 3 To check the material dependency of the parameter n the values of C were also re-calculated for keeping the value ofn constant ie the same as the mean value obtained in the laboratory tests These values are also presented in the same table From examination of the product size distribution predictions by the two approaches it was found that constant n caused considerable detoriation in the quality of the fit
Table 3 C and n values calculated from Mogul plant data
Ball Mill 1 Ball Mill 2 C 1455xlO-3 9l23xlO-4
n 092427 098457 n=07316 C 33 12x 1 0-3 2779xlO-3
Although it may be argued that the difference in the value of n may be due to the simplification of residence time distribution by plug flow assumption or wet and closed circuited nature of plant scale grinding this may also be due to increased mill diameter An assumption that the fines would have shorter retention times than the coarse fractions could decrease the value n to the value obtained in the laboratory Nevertheless it is believed that the present data is insufficient to draw firm conclusion particularly in terms of scale up of parameter n The parameter C on the other hand had a value within the range of C values obtained in the laboratory tests indicating the value of C may be predicted by laboratory test conducted under carefully chosen grinding conditions Obviously for the development of such procedure similar experimental studies with several other ores are needed Therefore the results
presented here are only indicative rather than conclusive A comparison of plant parameter with those
obtained from test conducted under the test conditions indicated that the plant grinding was slightly less inefficient Though the linearity ofBond results is rather questionable Since Bond test is used for the industrial mills similar efficiency could have been expected However it may still be argued that the Bond test conditions could simulate the actual grinding conditions if it is considered that some inefficiency is expected due to diameter and feed size factors etc
4 CONCLUSIONS
The result indicated that is reasonable to assume that n is a material dependent parameter although this should be verified for different materials and grinding conditions Then parameter C can be used as an indicator of for grinding efficiency Of the variables studied it is found that there exist
optimum values for ball size and ball filling for efficient grinding On the other hand the efficiency increased asymptotically with increasing critical speed and decreased with decreasing powder filling
The result also indicate that it may be possible to predict the model parameters of plant scale grinding by conducting laboratory tests suitably selected for this purpose Although it is rather arguable this point deserves further investigations by carrying out grinding tests under similar conditions using several different ores
REFERENCES
Austin L G Klimpel R R Luckie P T 1984 Process engineering of size reduction ball milling AIME Pub NY 584p
Bergstrom BH 1985 Crushability and grindability SME Mineral Processing Handbook Ed NL Weiss New York vol2 in Section30 Sampling and Testing pp65-68
Ersaym S Sonmez B Ergiin ~L Aksam B Erkal 1 F 1993 Simulation of the grinding circuit at Giimiisectkoy silver plant Turkey Trans fUM Sect C January-April vo1102 C32-38
Fidan B 1990 Grinding characteristics of Kilre copper ore MSc Thesis (unpublished) Mining Eng Dept Middle East Technical University 168p
537
Finch A J Ramirez Castro J 1981 Modeling of mineral size reduction in the closed circuit ball mill at the Pine Point Mines Concentrator Int J aMin Proc 861-78
Herbst J A Fuerstenau D W 1980 Scale up for continuous grinding mill design populance balance models Int J Min Proc 7 1-31
Laplante AR Finch J and del-Villar R 1987 Simplification of grinding equation for plant simulation TransIMM vol96 CI08-I12
Lianxiang L Bingchen C Liu Q 1988 A Study of grinding kinetics and its applications to the choice and calculation of ball media XVI Int Min Proc Congr Edited by Forssberg E Elsevier Science Publishers B v Amsterdam Netherlands 245-256
Lynch A J 1977 Mineral crushing and grinding circuits their simulation optimization design and control Elsevier Scientific Publishing Co Amsterdam 340p
Lynch AI Whiten WJ Narayanan SS 1986 Ball mill modelstheir evaluation and present status in Advances in Mineral Processing edP Somasundaran AlME Littleton Colorado p48shy66
Olsen TO and Krogh SR 1972 Mathematical model of grinding at different conditions in ball mills Trans SMFAIME p453-457
Ramirez Castro J and Finch A J and 1980 Simulation of a grinding circuit change to reduce lead sliming CIMM Bull vol 73 pp132-139
538
ball filling 875 of critical speed 18 powder filling and 254 cm diameter balls
After each grinding period the whole load in the mill was emptied and a representative sample was taken for sieve analysis Then the material retained on the screens were combined with the rest of mill discharge as a feed for next grinding period After sieve analysis model parameters cumulative basis kinetic model were determined
For the purposes of comparison and scale-up another test was carried out under the standard Bond test conditions (Bergstrom 1985) and the two ball mill operating parallel in Mogul plant were sampled After mass balancing and adjusting the raw data based on plug flow assumption the retention time of 591 minute was estimated for ball mill From the combined feed (fresh feed + cyclone undersize) and mill discharge size distributions the rate parameters were calculated by using the estimated retention time The model parameters C and n of the mills were then determined
Cumulative basis kinetic model
(1)Wi(t) = Wi(O)middote-kit ki=CxP (2)
cumulative oversize of size i after minute grinding
Wi(O)
Wi(t)
cumulative oversize of size i in the feed ki rate parameter (min-I) t grinding time (min) Xi particle size (Ilm) Cn model parameters
From size distribution data first rate parameter k of each size fraction and then using Equation 2 C and n parameters were determined A non-linear regression technique was used for the calculation of model parameters
3 RESULTS AND DISCUSSION
After the experiments the experimental data showed a quite good fit to both Equation 1 and 2 except the results of the test conducted with 127 cm balls This may be due to insufficient energy exerted to the particles from 127 cm diameter balls Effects of operational conditions on k parameters are shown at Figure 1-4
As can be seen from the figures the effects of operating conditions on rate parameter k were more pronounced at coarser size range Rate parameters decreased with increasing powder filling while they increased asymptotically with increasing mill speed It also appeared that there exist optimum ball size and ball filling for obtaining maximum efficiency
534
---+--425flIl 08 --IJ-300flIl
88GiSpeed middotmiddotmiddotmiddotjmiddotmiddotmiddotmiddot212flIl06 _ __ 150flIl
x04 ___ l06flIl
middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot75flIl02 I ~ltoo~ I --+--53flIl o I I I
5 10 15 20 25 30 35 40 powler filling
Figure 1 Effect of powder filling on rate parameters of different sizes (88 critical speed)
02 ----------- --+-425flIl
oos
44Gi~ ---- - lOOflIl 015 ~
~~ l 212flIl
OJ
~~ _middot-X-middot-lSOfID
--I--l~~~~ ---e--75flIl
--+-53flIl
O+I---+----r--~
15 30 45 ro jXJoOOrfilling
Figure 2 Effect of powder filling on rate parameters of different sizes (44 critical speed)
05
1 ---+-- 425fJ11
--IJ-300fJ11 04 ------~--
j 212fJ11 ~--03 ~l
-middot-)E-middot_middot 150fJ11I E--------_x
__- 106fJ11 ~=CCCl emiddotmiddot75fJ11
o -+-53fJ11
15 25 35 45 ballfilling
Figure 3 Effect of ball filling on rate parameters of different sizes
c 6 g
a ~
001 lts I I
Immmmll IOlg~~~
10 particle sire (pm) 1000
Figure 6 Relationship between rate parameter and particle size for different conditions
100
Candn
o constantnfmiddot bull _J II70 ]60 ]50 1
140 30 1
20 10
o I
~V
~ W~SWlln~
I~ 0
0 20 40 60 80 1~1 observed cumulative ovemizc(
Figure 7 Comparison of fit obtained by using C and n and constant n
001 poMerlilling
o~ ~ bull ~scil 0006 bull 4375Csi
----~s CS 0004 Imiddot middotmiddot 4375 cs1
bull0002 ~ --bull o +-1---+---+----1------+-----1----1
o 10 20 30 40 50 00
Figure 8 Effect of powder filling on C parameter
536
~~ r----------------------------- 0007
~006
0005 ~-~ 0004
0003
0002 +--------+--------+------------1 15 25 35 45
Figure 9 Effect ofball filling on C parameter
~006
ball size 0005
10004
I ~003
I 0002
I0001 1 2 3
Figure 10 Effect ofbaIl size on C parameter
0005 ~I
0004
Trill speed 0003
OOOZ+I----~------~----+_----~----~
40 50 00 70 80
Figure 11 Effect ofmill speed on C parameter
4
05
Q4
03 - Q2
Ql
o +---~~~~---+---+--~ 1 U 2 ~ 3 ~ 4
w~
Figure 4 Effect of ball size on rate parameters of different sizes
-+-~ 425flIl 04
---middot300flIl
middotmiddotmiddotmiddotmiddottmiddotmiddotmiddotmiddotmiddotmiddot 212flIl
-_- 150flIl 02
03 ~--------
t~ _-- l06flIl
bullbullbull 15flIlOJ -middot+-53flIl
I I I Io 40 50 ro 70 ID lt0
niIlspeed
Figure 5 Effect of mill speed on rate parameters of different sizes
The model parameters calculated from rate parameters are presented at Table 2 A close examination of the values of n obtained under different laboratory conditions (excluding the test under Bond conditions) revealed that n was almost constant and the changes in conditions were mainly reflected on the values of parameter C This can also be seen from Figure 6 where the slopes of lines were nearly the same indicating constant n There was only one test at which n showed large deviation This was obtained when ball size of 127 cm was used As explained earlier inefficient grinding with this size of balls was observed Excluding this test it may be assumed that n could be considered as a material dependent parameter This would mean that the changes on grinding conditions will be reflected on the values of parameter C
535
Table 2 C and n parameters ofthe tests
Cx163 n powder filling () at 875 cs 89 758 078 178 559 070 356 lAO 074 at 43 cs 178 254 071 356 283 064 572 142 064 mill speed ( of cs) 438 254 071 625 221 085 875 559 070 ball filling () 179 219 077 265 412 079 441 559 070 ball size (cm) 127 012 040 254 559 070 381 250 081 Bond Conditions 324 074
To check the validity of this assumption the mean value of n (excluding the extreme result) was first used in the re-calculations ofparameter C Then the predicted size distributions were calculated by using the new model parameters As is shown in Figure 7 both approaches provided similar fit to the data Thereforeit was concluded that under the laboratory conditions studied the model parameter n was reasonably constant indicating that it may be a material dependent parameter
The relationships between the variables studied and the re-calculated (for n=07316) C parameter are presented in Figure 8-11 The trends were similar to those obtained for the rate parameters Of the variables studied the most significant effect was observed for the powder filling Le C decreased sharply with its increase However the curvature gives the impression that further increase would not have such sharp effect Perhaps it would stay almost constant after a certain level Nevertheless this implies that plant scale grinding is performed at inefficient conditions since mills are normally operated at 100-120 powder filling On the other hand the optimum obtained for ball filling is around the normal operating conditions and an expected relationship was observed between C and critical speed It may be suggested the optimum ball size could vary with the feed size ofthe material
Although the plot of log k vs log particle size showed relatively larger deviation from linearity for the test conducted under Bond conditions (thick solid line in Figure 6) the value of n calculated by non-linear regression was almost the same as the mean values ofthe other laboratory tests Since there is only one set of data it would be very much arguable to suggest that the mixture of ball might be the cause of the deviation from linearity From the examination of rate parameters of individual sizes It appears that deviation mainly occurs due to inefficient grinding in fine sizes Although not so much pronounced similar deviations are also discernible in the parameters of the other tests The value of C obtained from the test under Bond
conditions is within the range studied but it did not exactly correspond to the most similar test conditions with one size balls It is slightly lower than the expected The model parameters calculated from the plant data is given Table 3 To check the material dependency of the parameter n the values of C were also re-calculated for keeping the value ofn constant ie the same as the mean value obtained in the laboratory tests These values are also presented in the same table From examination of the product size distribution predictions by the two approaches it was found that constant n caused considerable detoriation in the quality of the fit
Table 3 C and n values calculated from Mogul plant data
Ball Mill 1 Ball Mill 2 C 1455xlO-3 9l23xlO-4
n 092427 098457 n=07316 C 33 12x 1 0-3 2779xlO-3
Although it may be argued that the difference in the value of n may be due to the simplification of residence time distribution by plug flow assumption or wet and closed circuited nature of plant scale grinding this may also be due to increased mill diameter An assumption that the fines would have shorter retention times than the coarse fractions could decrease the value n to the value obtained in the laboratory Nevertheless it is believed that the present data is insufficient to draw firm conclusion particularly in terms of scale up of parameter n The parameter C on the other hand had a value within the range of C values obtained in the laboratory tests indicating the value of C may be predicted by laboratory test conducted under carefully chosen grinding conditions Obviously for the development of such procedure similar experimental studies with several other ores are needed Therefore the results
presented here are only indicative rather than conclusive A comparison of plant parameter with those
obtained from test conducted under the test conditions indicated that the plant grinding was slightly less inefficient Though the linearity ofBond results is rather questionable Since Bond test is used for the industrial mills similar efficiency could have been expected However it may still be argued that the Bond test conditions could simulate the actual grinding conditions if it is considered that some inefficiency is expected due to diameter and feed size factors etc
4 CONCLUSIONS
The result indicated that is reasonable to assume that n is a material dependent parameter although this should be verified for different materials and grinding conditions Then parameter C can be used as an indicator of for grinding efficiency Of the variables studied it is found that there exist
optimum values for ball size and ball filling for efficient grinding On the other hand the efficiency increased asymptotically with increasing critical speed and decreased with decreasing powder filling
The result also indicate that it may be possible to predict the model parameters of plant scale grinding by conducting laboratory tests suitably selected for this purpose Although it is rather arguable this point deserves further investigations by carrying out grinding tests under similar conditions using several different ores
REFERENCES
Austin L G Klimpel R R Luckie P T 1984 Process engineering of size reduction ball milling AIME Pub NY 584p
Bergstrom BH 1985 Crushability and grindability SME Mineral Processing Handbook Ed NL Weiss New York vol2 in Section30 Sampling and Testing pp65-68
Ersaym S Sonmez B Ergiin ~L Aksam B Erkal 1 F 1993 Simulation of the grinding circuit at Giimiisectkoy silver plant Turkey Trans fUM Sect C January-April vo1102 C32-38
Fidan B 1990 Grinding characteristics of Kilre copper ore MSc Thesis (unpublished) Mining Eng Dept Middle East Technical University 168p
537
Finch A J Ramirez Castro J 1981 Modeling of mineral size reduction in the closed circuit ball mill at the Pine Point Mines Concentrator Int J aMin Proc 861-78
Herbst J A Fuerstenau D W 1980 Scale up for continuous grinding mill design populance balance models Int J Min Proc 7 1-31
Laplante AR Finch J and del-Villar R 1987 Simplification of grinding equation for plant simulation TransIMM vol96 CI08-I12
Lianxiang L Bingchen C Liu Q 1988 A Study of grinding kinetics and its applications to the choice and calculation of ball media XVI Int Min Proc Congr Edited by Forssberg E Elsevier Science Publishers B v Amsterdam Netherlands 245-256
Lynch A J 1977 Mineral crushing and grinding circuits their simulation optimization design and control Elsevier Scientific Publishing Co Amsterdam 340p
Lynch AI Whiten WJ Narayanan SS 1986 Ball mill modelstheir evaluation and present status in Advances in Mineral Processing edP Somasundaran AlME Littleton Colorado p48shy66
Olsen TO and Krogh SR 1972 Mathematical model of grinding at different conditions in ball mills Trans SMFAIME p453-457
Ramirez Castro J and Finch A J and 1980 Simulation of a grinding circuit change to reduce lead sliming CIMM Bull vol 73 pp132-139
538
c 6 g
a ~
001 lts I I
Immmmll IOlg~~~
10 particle sire (pm) 1000
Figure 6 Relationship between rate parameter and particle size for different conditions
100
Candn
o constantnfmiddot bull _J II70 ]60 ]50 1
140 30 1
20 10
o I
~V
~ W~SWlln~
I~ 0
0 20 40 60 80 1~1 observed cumulative ovemizc(
Figure 7 Comparison of fit obtained by using C and n and constant n
001 poMerlilling
o~ ~ bull ~scil 0006 bull 4375Csi
----~s CS 0004 Imiddot middotmiddot 4375 cs1
bull0002 ~ --bull o +-1---+---+----1------+-----1----1
o 10 20 30 40 50 00
Figure 8 Effect of powder filling on C parameter
536
~~ r----------------------------- 0007
~006
0005 ~-~ 0004
0003
0002 +--------+--------+------------1 15 25 35 45
Figure 9 Effect ofball filling on C parameter
~006
ball size 0005
10004
I ~003
I 0002
I0001 1 2 3
Figure 10 Effect ofbaIl size on C parameter
0005 ~I
0004
Trill speed 0003
OOOZ+I----~------~----+_----~----~
40 50 00 70 80
Figure 11 Effect ofmill speed on C parameter
4
05
Q4
03 - Q2
Ql
o +---~~~~---+---+--~ 1 U 2 ~ 3 ~ 4
w~
Figure 4 Effect of ball size on rate parameters of different sizes
-+-~ 425flIl 04
---middot300flIl
middotmiddotmiddotmiddotmiddottmiddotmiddotmiddotmiddotmiddotmiddot 212flIl
-_- 150flIl 02
03 ~--------
t~ _-- l06flIl
bullbullbull 15flIlOJ -middot+-53flIl
I I I Io 40 50 ro 70 ID lt0
niIlspeed
Figure 5 Effect of mill speed on rate parameters of different sizes
The model parameters calculated from rate parameters are presented at Table 2 A close examination of the values of n obtained under different laboratory conditions (excluding the test under Bond conditions) revealed that n was almost constant and the changes in conditions were mainly reflected on the values of parameter C This can also be seen from Figure 6 where the slopes of lines were nearly the same indicating constant n There was only one test at which n showed large deviation This was obtained when ball size of 127 cm was used As explained earlier inefficient grinding with this size of balls was observed Excluding this test it may be assumed that n could be considered as a material dependent parameter This would mean that the changes on grinding conditions will be reflected on the values of parameter C
535
Table 2 C and n parameters ofthe tests
Cx163 n powder filling () at 875 cs 89 758 078 178 559 070 356 lAO 074 at 43 cs 178 254 071 356 283 064 572 142 064 mill speed ( of cs) 438 254 071 625 221 085 875 559 070 ball filling () 179 219 077 265 412 079 441 559 070 ball size (cm) 127 012 040 254 559 070 381 250 081 Bond Conditions 324 074
To check the validity of this assumption the mean value of n (excluding the extreme result) was first used in the re-calculations ofparameter C Then the predicted size distributions were calculated by using the new model parameters As is shown in Figure 7 both approaches provided similar fit to the data Thereforeit was concluded that under the laboratory conditions studied the model parameter n was reasonably constant indicating that it may be a material dependent parameter
The relationships between the variables studied and the re-calculated (for n=07316) C parameter are presented in Figure 8-11 The trends were similar to those obtained for the rate parameters Of the variables studied the most significant effect was observed for the powder filling Le C decreased sharply with its increase However the curvature gives the impression that further increase would not have such sharp effect Perhaps it would stay almost constant after a certain level Nevertheless this implies that plant scale grinding is performed at inefficient conditions since mills are normally operated at 100-120 powder filling On the other hand the optimum obtained for ball filling is around the normal operating conditions and an expected relationship was observed between C and critical speed It may be suggested the optimum ball size could vary with the feed size ofthe material
Although the plot of log k vs log particle size showed relatively larger deviation from linearity for the test conducted under Bond conditions (thick solid line in Figure 6) the value of n calculated by non-linear regression was almost the same as the mean values ofthe other laboratory tests Since there is only one set of data it would be very much arguable to suggest that the mixture of ball might be the cause of the deviation from linearity From the examination of rate parameters of individual sizes It appears that deviation mainly occurs due to inefficient grinding in fine sizes Although not so much pronounced similar deviations are also discernible in the parameters of the other tests The value of C obtained from the test under Bond
conditions is within the range studied but it did not exactly correspond to the most similar test conditions with one size balls It is slightly lower than the expected The model parameters calculated from the plant data is given Table 3 To check the material dependency of the parameter n the values of C were also re-calculated for keeping the value ofn constant ie the same as the mean value obtained in the laboratory tests These values are also presented in the same table From examination of the product size distribution predictions by the two approaches it was found that constant n caused considerable detoriation in the quality of the fit
Table 3 C and n values calculated from Mogul plant data
Ball Mill 1 Ball Mill 2 C 1455xlO-3 9l23xlO-4
n 092427 098457 n=07316 C 33 12x 1 0-3 2779xlO-3
Although it may be argued that the difference in the value of n may be due to the simplification of residence time distribution by plug flow assumption or wet and closed circuited nature of plant scale grinding this may also be due to increased mill diameter An assumption that the fines would have shorter retention times than the coarse fractions could decrease the value n to the value obtained in the laboratory Nevertheless it is believed that the present data is insufficient to draw firm conclusion particularly in terms of scale up of parameter n The parameter C on the other hand had a value within the range of C values obtained in the laboratory tests indicating the value of C may be predicted by laboratory test conducted under carefully chosen grinding conditions Obviously for the development of such procedure similar experimental studies with several other ores are needed Therefore the results
presented here are only indicative rather than conclusive A comparison of plant parameter with those
obtained from test conducted under the test conditions indicated that the plant grinding was slightly less inefficient Though the linearity ofBond results is rather questionable Since Bond test is used for the industrial mills similar efficiency could have been expected However it may still be argued that the Bond test conditions could simulate the actual grinding conditions if it is considered that some inefficiency is expected due to diameter and feed size factors etc
4 CONCLUSIONS
The result indicated that is reasonable to assume that n is a material dependent parameter although this should be verified for different materials and grinding conditions Then parameter C can be used as an indicator of for grinding efficiency Of the variables studied it is found that there exist
optimum values for ball size and ball filling for efficient grinding On the other hand the efficiency increased asymptotically with increasing critical speed and decreased with decreasing powder filling
The result also indicate that it may be possible to predict the model parameters of plant scale grinding by conducting laboratory tests suitably selected for this purpose Although it is rather arguable this point deserves further investigations by carrying out grinding tests under similar conditions using several different ores
REFERENCES
Austin L G Klimpel R R Luckie P T 1984 Process engineering of size reduction ball milling AIME Pub NY 584p
Bergstrom BH 1985 Crushability and grindability SME Mineral Processing Handbook Ed NL Weiss New York vol2 in Section30 Sampling and Testing pp65-68
Ersaym S Sonmez B Ergiin ~L Aksam B Erkal 1 F 1993 Simulation of the grinding circuit at Giimiisectkoy silver plant Turkey Trans fUM Sect C January-April vo1102 C32-38
Fidan B 1990 Grinding characteristics of Kilre copper ore MSc Thesis (unpublished) Mining Eng Dept Middle East Technical University 168p
537
Finch A J Ramirez Castro J 1981 Modeling of mineral size reduction in the closed circuit ball mill at the Pine Point Mines Concentrator Int J aMin Proc 861-78
Herbst J A Fuerstenau D W 1980 Scale up for continuous grinding mill design populance balance models Int J Min Proc 7 1-31
Laplante AR Finch J and del-Villar R 1987 Simplification of grinding equation for plant simulation TransIMM vol96 CI08-I12
Lianxiang L Bingchen C Liu Q 1988 A Study of grinding kinetics and its applications to the choice and calculation of ball media XVI Int Min Proc Congr Edited by Forssberg E Elsevier Science Publishers B v Amsterdam Netherlands 245-256
Lynch A J 1977 Mineral crushing and grinding circuits their simulation optimization design and control Elsevier Scientific Publishing Co Amsterdam 340p
Lynch AI Whiten WJ Narayanan SS 1986 Ball mill modelstheir evaluation and present status in Advances in Mineral Processing edP Somasundaran AlME Littleton Colorado p48shy66
Olsen TO and Krogh SR 1972 Mathematical model of grinding at different conditions in ball mills Trans SMFAIME p453-457
Ramirez Castro J and Finch A J and 1980 Simulation of a grinding circuit change to reduce lead sliming CIMM Bull vol 73 pp132-139
538
05
Q4
03 - Q2
Ql
o +---~~~~---+---+--~ 1 U 2 ~ 3 ~ 4
w~
Figure 4 Effect of ball size on rate parameters of different sizes
-+-~ 425flIl 04
---middot300flIl
middotmiddotmiddotmiddotmiddottmiddotmiddotmiddotmiddotmiddotmiddot 212flIl
-_- 150flIl 02
03 ~--------
t~ _-- l06flIl
bullbullbull 15flIlOJ -middot+-53flIl
I I I Io 40 50 ro 70 ID lt0
niIlspeed
Figure 5 Effect of mill speed on rate parameters of different sizes
The model parameters calculated from rate parameters are presented at Table 2 A close examination of the values of n obtained under different laboratory conditions (excluding the test under Bond conditions) revealed that n was almost constant and the changes in conditions were mainly reflected on the values of parameter C This can also be seen from Figure 6 where the slopes of lines were nearly the same indicating constant n There was only one test at which n showed large deviation This was obtained when ball size of 127 cm was used As explained earlier inefficient grinding with this size of balls was observed Excluding this test it may be assumed that n could be considered as a material dependent parameter This would mean that the changes on grinding conditions will be reflected on the values of parameter C
535
Table 2 C and n parameters ofthe tests
Cx163 n powder filling () at 875 cs 89 758 078 178 559 070 356 lAO 074 at 43 cs 178 254 071 356 283 064 572 142 064 mill speed ( of cs) 438 254 071 625 221 085 875 559 070 ball filling () 179 219 077 265 412 079 441 559 070 ball size (cm) 127 012 040 254 559 070 381 250 081 Bond Conditions 324 074
To check the validity of this assumption the mean value of n (excluding the extreme result) was first used in the re-calculations ofparameter C Then the predicted size distributions were calculated by using the new model parameters As is shown in Figure 7 both approaches provided similar fit to the data Thereforeit was concluded that under the laboratory conditions studied the model parameter n was reasonably constant indicating that it may be a material dependent parameter
The relationships between the variables studied and the re-calculated (for n=07316) C parameter are presented in Figure 8-11 The trends were similar to those obtained for the rate parameters Of the variables studied the most significant effect was observed for the powder filling Le C decreased sharply with its increase However the curvature gives the impression that further increase would not have such sharp effect Perhaps it would stay almost constant after a certain level Nevertheless this implies that plant scale grinding is performed at inefficient conditions since mills are normally operated at 100-120 powder filling On the other hand the optimum obtained for ball filling is around the normal operating conditions and an expected relationship was observed between C and critical speed It may be suggested the optimum ball size could vary with the feed size ofthe material
Although the plot of log k vs log particle size showed relatively larger deviation from linearity for the test conducted under Bond conditions (thick solid line in Figure 6) the value of n calculated by non-linear regression was almost the same as the mean values ofthe other laboratory tests Since there is only one set of data it would be very much arguable to suggest that the mixture of ball might be the cause of the deviation from linearity From the examination of rate parameters of individual sizes It appears that deviation mainly occurs due to inefficient grinding in fine sizes Although not so much pronounced similar deviations are also discernible in the parameters of the other tests The value of C obtained from the test under Bond
conditions is within the range studied but it did not exactly correspond to the most similar test conditions with one size balls It is slightly lower than the expected The model parameters calculated from the plant data is given Table 3 To check the material dependency of the parameter n the values of C were also re-calculated for keeping the value ofn constant ie the same as the mean value obtained in the laboratory tests These values are also presented in the same table From examination of the product size distribution predictions by the two approaches it was found that constant n caused considerable detoriation in the quality of the fit
Table 3 C and n values calculated from Mogul plant data
Ball Mill 1 Ball Mill 2 C 1455xlO-3 9l23xlO-4
n 092427 098457 n=07316 C 33 12x 1 0-3 2779xlO-3
Although it may be argued that the difference in the value of n may be due to the simplification of residence time distribution by plug flow assumption or wet and closed circuited nature of plant scale grinding this may also be due to increased mill diameter An assumption that the fines would have shorter retention times than the coarse fractions could decrease the value n to the value obtained in the laboratory Nevertheless it is believed that the present data is insufficient to draw firm conclusion particularly in terms of scale up of parameter n The parameter C on the other hand had a value within the range of C values obtained in the laboratory tests indicating the value of C may be predicted by laboratory test conducted under carefully chosen grinding conditions Obviously for the development of such procedure similar experimental studies with several other ores are needed Therefore the results
presented here are only indicative rather than conclusive A comparison of plant parameter with those
obtained from test conducted under the test conditions indicated that the plant grinding was slightly less inefficient Though the linearity ofBond results is rather questionable Since Bond test is used for the industrial mills similar efficiency could have been expected However it may still be argued that the Bond test conditions could simulate the actual grinding conditions if it is considered that some inefficiency is expected due to diameter and feed size factors etc
4 CONCLUSIONS
The result indicated that is reasonable to assume that n is a material dependent parameter although this should be verified for different materials and grinding conditions Then parameter C can be used as an indicator of for grinding efficiency Of the variables studied it is found that there exist
optimum values for ball size and ball filling for efficient grinding On the other hand the efficiency increased asymptotically with increasing critical speed and decreased with decreasing powder filling
The result also indicate that it may be possible to predict the model parameters of plant scale grinding by conducting laboratory tests suitably selected for this purpose Although it is rather arguable this point deserves further investigations by carrying out grinding tests under similar conditions using several different ores
REFERENCES
Austin L G Klimpel R R Luckie P T 1984 Process engineering of size reduction ball milling AIME Pub NY 584p
Bergstrom BH 1985 Crushability and grindability SME Mineral Processing Handbook Ed NL Weiss New York vol2 in Section30 Sampling and Testing pp65-68
Ersaym S Sonmez B Ergiin ~L Aksam B Erkal 1 F 1993 Simulation of the grinding circuit at Giimiisectkoy silver plant Turkey Trans fUM Sect C January-April vo1102 C32-38
Fidan B 1990 Grinding characteristics of Kilre copper ore MSc Thesis (unpublished) Mining Eng Dept Middle East Technical University 168p
537
Finch A J Ramirez Castro J 1981 Modeling of mineral size reduction in the closed circuit ball mill at the Pine Point Mines Concentrator Int J aMin Proc 861-78
Herbst J A Fuerstenau D W 1980 Scale up for continuous grinding mill design populance balance models Int J Min Proc 7 1-31
Laplante AR Finch J and del-Villar R 1987 Simplification of grinding equation for plant simulation TransIMM vol96 CI08-I12
Lianxiang L Bingchen C Liu Q 1988 A Study of grinding kinetics and its applications to the choice and calculation of ball media XVI Int Min Proc Congr Edited by Forssberg E Elsevier Science Publishers B v Amsterdam Netherlands 245-256
Lynch A J 1977 Mineral crushing and grinding circuits their simulation optimization design and control Elsevier Scientific Publishing Co Amsterdam 340p
Lynch AI Whiten WJ Narayanan SS 1986 Ball mill modelstheir evaluation and present status in Advances in Mineral Processing edP Somasundaran AlME Littleton Colorado p48shy66
Olsen TO and Krogh SR 1972 Mathematical model of grinding at different conditions in ball mills Trans SMFAIME p453-457
Ramirez Castro J and Finch A J and 1980 Simulation of a grinding circuit change to reduce lead sliming CIMM Bull vol 73 pp132-139
538
Although the plot of log k vs log particle size showed relatively larger deviation from linearity for the test conducted under Bond conditions (thick solid line in Figure 6) the value of n calculated by non-linear regression was almost the same as the mean values ofthe other laboratory tests Since there is only one set of data it would be very much arguable to suggest that the mixture of ball might be the cause of the deviation from linearity From the examination of rate parameters of individual sizes It appears that deviation mainly occurs due to inefficient grinding in fine sizes Although not so much pronounced similar deviations are also discernible in the parameters of the other tests The value of C obtained from the test under Bond
conditions is within the range studied but it did not exactly correspond to the most similar test conditions with one size balls It is slightly lower than the expected The model parameters calculated from the plant data is given Table 3 To check the material dependency of the parameter n the values of C were also re-calculated for keeping the value ofn constant ie the same as the mean value obtained in the laboratory tests These values are also presented in the same table From examination of the product size distribution predictions by the two approaches it was found that constant n caused considerable detoriation in the quality of the fit
Table 3 C and n values calculated from Mogul plant data
Ball Mill 1 Ball Mill 2 C 1455xlO-3 9l23xlO-4
n 092427 098457 n=07316 C 33 12x 1 0-3 2779xlO-3
Although it may be argued that the difference in the value of n may be due to the simplification of residence time distribution by plug flow assumption or wet and closed circuited nature of plant scale grinding this may also be due to increased mill diameter An assumption that the fines would have shorter retention times than the coarse fractions could decrease the value n to the value obtained in the laboratory Nevertheless it is believed that the present data is insufficient to draw firm conclusion particularly in terms of scale up of parameter n The parameter C on the other hand had a value within the range of C values obtained in the laboratory tests indicating the value of C may be predicted by laboratory test conducted under carefully chosen grinding conditions Obviously for the development of such procedure similar experimental studies with several other ores are needed Therefore the results
presented here are only indicative rather than conclusive A comparison of plant parameter with those
obtained from test conducted under the test conditions indicated that the plant grinding was slightly less inefficient Though the linearity ofBond results is rather questionable Since Bond test is used for the industrial mills similar efficiency could have been expected However it may still be argued that the Bond test conditions could simulate the actual grinding conditions if it is considered that some inefficiency is expected due to diameter and feed size factors etc
4 CONCLUSIONS
The result indicated that is reasonable to assume that n is a material dependent parameter although this should be verified for different materials and grinding conditions Then parameter C can be used as an indicator of for grinding efficiency Of the variables studied it is found that there exist
optimum values for ball size and ball filling for efficient grinding On the other hand the efficiency increased asymptotically with increasing critical speed and decreased with decreasing powder filling
The result also indicate that it may be possible to predict the model parameters of plant scale grinding by conducting laboratory tests suitably selected for this purpose Although it is rather arguable this point deserves further investigations by carrying out grinding tests under similar conditions using several different ores
REFERENCES
Austin L G Klimpel R R Luckie P T 1984 Process engineering of size reduction ball milling AIME Pub NY 584p
Bergstrom BH 1985 Crushability and grindability SME Mineral Processing Handbook Ed NL Weiss New York vol2 in Section30 Sampling and Testing pp65-68
Ersaym S Sonmez B Ergiin ~L Aksam B Erkal 1 F 1993 Simulation of the grinding circuit at Giimiisectkoy silver plant Turkey Trans fUM Sect C January-April vo1102 C32-38
Fidan B 1990 Grinding characteristics of Kilre copper ore MSc Thesis (unpublished) Mining Eng Dept Middle East Technical University 168p
537
Finch A J Ramirez Castro J 1981 Modeling of mineral size reduction in the closed circuit ball mill at the Pine Point Mines Concentrator Int J aMin Proc 861-78
Herbst J A Fuerstenau D W 1980 Scale up for continuous grinding mill design populance balance models Int J Min Proc 7 1-31
Laplante AR Finch J and del-Villar R 1987 Simplification of grinding equation for plant simulation TransIMM vol96 CI08-I12
Lianxiang L Bingchen C Liu Q 1988 A Study of grinding kinetics and its applications to the choice and calculation of ball media XVI Int Min Proc Congr Edited by Forssberg E Elsevier Science Publishers B v Amsterdam Netherlands 245-256
Lynch A J 1977 Mineral crushing and grinding circuits their simulation optimization design and control Elsevier Scientific Publishing Co Amsterdam 340p
Lynch AI Whiten WJ Narayanan SS 1986 Ball mill modelstheir evaluation and present status in Advances in Mineral Processing edP Somasundaran AlME Littleton Colorado p48shy66
Olsen TO and Krogh SR 1972 Mathematical model of grinding at different conditions in ball mills Trans SMFAIME p453-457
Ramirez Castro J and Finch A J and 1980 Simulation of a grinding circuit change to reduce lead sliming CIMM Bull vol 73 pp132-139
538
Finch A J Ramirez Castro J 1981 Modeling of mineral size reduction in the closed circuit ball mill at the Pine Point Mines Concentrator Int J aMin Proc 861-78
Herbst J A Fuerstenau D W 1980 Scale up for continuous grinding mill design populance balance models Int J Min Proc 7 1-31
Laplante AR Finch J and del-Villar R 1987 Simplification of grinding equation for plant simulation TransIMM vol96 CI08-I12
Lianxiang L Bingchen C Liu Q 1988 A Study of grinding kinetics and its applications to the choice and calculation of ball media XVI Int Min Proc Congr Edited by Forssberg E Elsevier Science Publishers B v Amsterdam Netherlands 245-256
Lynch A J 1977 Mineral crushing and grinding circuits their simulation optimization design and control Elsevier Scientific Publishing Co Amsterdam 340p
Lynch AI Whiten WJ Narayanan SS 1986 Ball mill modelstheir evaluation and present status in Advances in Mineral Processing edP Somasundaran AlME Littleton Colorado p48shy66
Olsen TO and Krogh SR 1972 Mathematical model of grinding at different conditions in ball mills Trans SMFAIME p453-457
Ramirez Castro J and Finch A J and 1980 Simulation of a grinding circuit change to reduce lead sliming CIMM Bull vol 73 pp132-139
538
